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K3 Surface

A K3 surface is a complex surface X whose underlying smooth manifold is a compact manifold, whose canonical bundle is trivial, and for which H^1(X,O_X)=0. Equivalently, it is a simply connected complex surface that is a compact manifold and has a nowhere-vanishing holomorphic differential form that is a 2-form. A complex K3 surface is a two-dimensional Calabi-Yau space and also a hyper-Kähler manifold.

The Betti numbers and Euler characteristic of every complex K3 surface are

(b_0,b_1,b_2,b_3,b_4)=(1,0,22,0,1), chi(X)=24.

Its Hodge numbers satisfy h^(2,0)=h^(0,2)=1 and h^(1,1)=20.

Every smooth quartic surface in complex projective space P^3(C) is a K3 surface. Other standard examples include double covers of the projective plane whose branch locus is a smooth sextic and Kummer surfaces, obtained by resolving the 16 singularities of the quotient space of a two-dimensional complex torus by the involution x|->-x.

OpenAI (2026c) announced a proof of the rational Hodge conjecture for every finite product of projective complex K3 surfaces. The claim was withdrawn on Oct. 7, 2026, together with the dependent Kuga-Satake correspondence claim, because a sign error invalidated a construction in the underlying paper on Weil classes (OpenAI 2026b). The announcement therefore no longer supplies a proof for these products.

See also

Calabi-Yau Space, Canonical Bundle, Generalized Hodge Conjecture, Hodge Conjecture, Hyper-Kähler Manifold, Irreducible Holomorphic Symplectic Manifold, Kuga-Satake Correspondence, Kummer Surface, Kuznetsov Component, Kuznetsov Rationality Conjecture, Projective Space

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References

Barth, W. P.; Peters, C. A.; and van de Ven, A. A. Compact Complex Surfaces. New York: Springer-Verlag, 1984.Huybrechts, D. Lectures on K3 Surfaces. Cambridge, England: Cambridge University Press, 2016. https://doi.org/10.1017/CBO9781316594193.OpenAI. "History." Oct. 7, 2026b. https://github.com/openai/math/blob/main/history.md.OpenAI. "The Rational Hodge Conjecture for Products of K3 Surfaces." Oct. 4, 2026c. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf.

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K3 Surface

Cite this as:

Weisstein, Eric W. "K3 Surface." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/K3Surface.html

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